Math ASCII Notation Demo

Mathematical content on Apronus.com is presented in Math ASCII Notation which can be properly displayed by all Web browsers because it uses only the basic set of characters found on all keyboards and in all fonts.

The purpose of these pages is to demonstrate the power of the Math ASCII Notation. In principle, it can be used to write mathematical content of any complexity. In practice, its limits can be seen when trying to write complicated formulas (containing, for example, variables with many indexes or multiple integrals).

Despite its limitations the Math ASCII Notation has much expressive power, as can be seen from browsing through these pages.

Let X be a set, and W c P(X).
(1) X :- W
(2) /\B,A:-W A n B :- W
Prove that W is a base for W*.
page 61 in gen top
Let X be a set, and W c P(X).
What does it mean that W is a topology?
1) O :- W
2) X :- W
3) /\B,A:-W [ B n A :- W ]
4) /\McW [ \\//M :- W ]
Let X be a set, and W c P(X).
If W is a topology, then W* = ???
W* = W
HINT:
From ( /\B,A:-W AnB:-W ) we conclude that W is a base for W*.
That is the subject matter of another item.
page 63 in gen top
Let X be a set. Let A,B c P(X).
Prove that A c B ==> A* c B*.
page 63 in gen top
Let W c P(X).
Let Wd denote the collection of all finite intersections of sets from W.
Consider the set Wd u {X}.
What interesting feature does it have?
It is a base for W*.
page 64 in gen top
Let W c P(X).
Is it always true that W* is the smallest topology containing W?
Yes.
Recall that W* always is a topology.
Recall that if AcBcP(X), then A*cB*.
Recall that if Y is a topology, then Y*=Y.
Put it all together.
page 64,66 in gen top
Let d be metric on |R.
Is it possible that (|R,d) is incomplete?
Yes.
d(x,y) = | arctan(x) - arctan(y) |
lim(n->oo) arctan(n) = PI/2
Hence the sequence {arctan(n)} is Cauchy with respect to the Euclidean metric.
Hence the sequence {n} is Cauchy with respect to d.
(1) X c |R is incomplete in |R with respect to the Euclidean metric.
(2) (X,d) is a complete metric space
(3) (|R,r) is an incomplete metric space
(4) (|R,r) and (X,d) are homeomorphic
Is this setup possible?
Let X = ] -PI/2 , PI/2 [.
Let d(y,x) = |tan(y) - tan(x)|.
Now (X,d) is a complete metric space.
Let r(x,y) = |arctan(x) - arctan(y)|.
Now (|R,r) is an incomplete metric space.
Let f : |R -> X, f(x) = arctan(x).
Now f is a homeomorphism.
If X is a set, what is an "outer measure" on X?
Y : P(X) -> [0,oo]
(1) Y(O) = 0
(2) A c B c X => Y(A) <= Y(B)
(3) Y is countably subadditive
If X is a set, what is a "pseudometric" on X?
d : X x X -> |R
(1) /\a:-X d(a,a) = 0
(2) /\a,b:-X d(a,b) = d(b,a)
(3) /\a,b,c:-X d(a,c) <= d(a,b) + d(b,c)
REMARK: Maybe it's convenient to allow +oo as a possible value d(x,y)=oo.